Given $ m \angle RPS = 2x + 112$, and $ m \angle QPR = 6x + 52$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
Answer: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since $\angle QPS$ is a straight angle, we know ${m\angle QPS = 180}$ Substitute in the expressions that were given for each measure: $ {6x + 52} + {2x + 112} = {180}$ Combine like terms: $ 8x + 164 = 180$ Subtract $164$ from both sides: $ 8x = 16$ Divide both sides by $8$ to find $x$ $ x = 2$ Substitute $2$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 6({2}) + 52$ Simplify: $ {m\angle QPR = 12 + 52}$ So ${m\angle QPR = 64}$.